Note on the (non-)smoothness of discrete time value functions in optimal stopping

Abstract

We consider the discrete time stopping problem \[ V(t,x) = \sup _{\tau }\mathbb {E}_{(t,x)}g(\tau , X_{\tau }), \] where $X$ is a random walk. It is well known that the value function $V$ is in general not smooth on the boundary of the continuation set $\partial C$. We show that under some conditions $V$ is not smooth in the interior of $C$ either. Even more, under some additional conditions we show that $V$ is not differentiable on a dense subset of $C$. As a guiding example we consider the Chow-Robbins game. We give evidence that $\partial C$ is not smooth and that $C$ is not convex, in the Chow-Robbins game and other examples.